Problem: You have found the following ages (in years) of 6 bears. The bears are randomly selected from the 28 bears at your local zoo: $ 46,\enspace 1,\enspace 20,\enspace 16,\enspace 12,\enspace 2$ Based on your sample, what is the average age of the bears? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 28 bears, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{888.04} + {231.04} + {14.44} + {0.04} + {17.64} + {201.64}} {{6 - 1}} $ $ {s^2} = \dfrac{{1352.84}}{{5}} = {270.57\text{ years}^2} $ We can estimate that the average bear at the zoo is 16.2 years old. There is a variance of 270.57 years $^2$.